Method and system for determining position of a body

ABSTRACT

A method for determining the position of a rotor with respect to a stator. The method includes generating offset N and Q signals when the rotor moves with respect to the stator and sampling the N and Q signals at a predetermined time interval. A section of a polar plot in which the N and Q signal sample lies is determined and compared to the section of a polar plot in which a previous N and Q signal sample lies. A long range rotor position is calculated by maintaining a count of the change of sections from previous N and Q signal samples. A sub-section rotor position for the N and Q signal sample is calculated and combined with the long range rotor position to create a linear position signal.

THE FIELD OF THE INVENTION

[0001] The present invention generally relates to sensing devices, andmore particularly to absolute position sensing devices with a highsensing resolution.

BACKGROUND OF THE INVENTION

[0002] Position sensors are used in a variety of devices to allowelectrical systems to sense the motion or position of moving objects andcomponents. Types of position sensors include, for example, analogpotentiometers, digital encoders and capacity sensors, among other typesof position sensors. These types of sensors may be used to track motionand position over a wide range of distances, and may have a similarlywide range of sensing resolutions. A position sensor's sensingresolution determines the incremental amount of motion or displacementdetectable by the sensor. For example, a higher sensing resolutionallows the censor to detect smaller increments of movement.

[0003] In some instances when tracking the movement and position of acomponent, it may be necessary or desirable to have a very high sensingresolution relative to the distance over which the position must betracked. For example, one possible use of high resolution positionsensors may be seen in the U.S. Pat. No. 5,557,596 to Gibson et al., inwhich an ultra high-density storage device is described and claimed. Theultra high-density storage device of Gibson et al. uses field emitterswhich generate electron beam currents. The electron beam currents writeinformation onto storage areas of a storage medium. The storage mediumis positioned on a movable rotor. The rotor is moved by one or moremicromovers with respect to the field emitters so that the emitters canwrite and access information at a number of storage areas on the storagemedium.

[0004] To properly control the micromovers for high-resolution devicessuch as the high-density storage device of Gibson et al., a positionsensor capable of accurately indicating the relative position of therotor with respect to the emitter wafer is needed. The sensor mustindicate the position of the rotor over the entire stroke of themicromover (perhaps on the order of 50 to 100 μm), yet supply resolutiondown to a small percentage of a track width (perhaps on the order of0.01 to 0.10 μm).

SUMMARY OF THE INVENTION

[0005] A method for determining the position of a rotor with respect toa stator is described herein. The method includes generating offset Nand Q signals when the rotor moves with respect to the stator andsampling the N and Q signals at a predetermined time interval. A sectionof a polar plot in which the N and Q signal sample lies is determinedand compared to the section of a polar plot in which a previous N and Qsignal sample lies. A long range rotor position is calculated bymaintaining a count of the change of sections from previous N and Qsignal samples. A sub-section rotor position for the N and Q signalsample is calculated and combined with the long range rotor position tocreate a linear position signal.

BRIEF DESCRIPTION OF THE DRAWINGS

[0006]FIG. 1 is a perspective view of a generalized two plate capacitor.

[0007]FIG. 2 is a perspective view of a generalized multi-platecapacitor.

[0008]FIG. 3 is an illustration of two sets of capacitive platesarranged such that when one set of plates reaches a point of minimumsensitivity, the other set of plates have maximum sensitivity.

[0009]FIG. 4 is a flow chart illustrating the system and method of oneembodiment of the invention.

[0010]FIG. 5 is a plot of an illustrative set of periodically varyingsignals obtained from two sets of capacitive plates arranged as in FIG.3.

[0011]FIG. 6 is a polar plot of the periodically varying signals shownin FIG. 5.

[0012]FIG. 7 is an NQ combiner table used in one embodiment of theinvention.

[0013]FIG. 8 is a graph illustrating the creation of a linear positionsignal using the system and method of one embodiment of the invention,prior to quantizing the signal data.

[0014]FIG. 9 is a graph illustrating the creation of a linear positionsignal like using the system and method of one embodiment of theinvention, after quantizing the signal data

[0015]FIG. 10 is a graph illustrating the effect of high frequencymodulation of the signal data.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0016] In the following detailed description of the preferredembodiments, reference is made to the accompanying drawings which form apart hereof, and in which is shown by way of illustration specificembodiments in which the invention may be practiced. It is to beunderstood that other embodiments may be utilized and structural orlogical changes may be made without departing from the scope of thepresent invention. The following detailed description, therefore, is notto be taken in a limiting sense, and the scope of the present inventionis defined by the appended claims.

[0017] A calibrated position sensor based on a capacitance measurementcan be used as a suitable high resolution encoder. Capacitive positionsensors typically detect changes in position by measuring capacitancebetween two relative moving pieces or components. That is, the charge ofthe capacitor is measured and used to calculate a relative positionbetween the two components.

[0018] In a device like that described in Gibson et al., the capacitiveplates can be positioned such that they do not consume areas preferredto be used for the storage medium and/or signal traces. For example, thein-plane or X-Y position of the rotor can be determined by mountingcapacitor plates on the bottom surface of the rotor (opposite thestorage medium) and on the device's stationary frame under the rotor. Ofcourse, any other suitable location may be used for mounting thecapacitive plates. As the rotor moves, the capacitive plates on therotor move laterally with respect to the capacitive plates on the frame(i.e., the stator) to cause the area of overlap between the plates tovary as the rotor position varies.

[0019]FIG. 1 is a perspective view of a generalized capacitor 10. Twoparallel electrically conductive plates 12 and 14 are separated by adistance d and have a dielectric material 16 between them, forming acapacitor whose capacitance value depends on the physical separation ofthe plates 12 and 14, the dielectric constant of the material betweenthe plates, and the area of the plates 12, 14. If the plates 12, 14 areof unequal area or are shifted relative to each other while parallel,the capacitance is determined by the area of one plate projected ontothe other when viewing the plates at a direction perpendicular to theplanes of the plates 12, 14, i.e., the “overlap” area A between theplates. The capacitance is given by the equation:

C=AK/d

[0020] where A is the overlap area, K is the dielectric constant of thematerial 16 between the plates, and d is the plate separation. Avariable capacitor thus provides a variable capacitance based on thesize of the overlap area A, where the plates can be shifted relative toeach other to change the size of area A, as indicated by arrows 18. Thedielectric material 16 between plates 12, 14 may be any suitabledielectric material, including air.

[0021] To achieve high sensitivity in terms of capacitance per unit ofmovement (on the order of femtoFarad per micrometer (fF/μm) of lateralmovement), multiple parallel conductive plates 12′, 14′ are commonlyemployed, such as illustrated in FIG. 2. In this case, the capacitanceis given by the equation:

C=(n−1)AK/d

[0022] where n is the number of plates, A is the total overlap area ofthe plates, K is the dielectric constant of the material between theplates, and d is the plate separation. The plate arrangement of FIG. 2will result in a capacitance that varies in a periodic or cyclic fashionwith a linear change in position of the rotor (comprised of plates 14′)as the plates are shifted relative to each other as indicated by arrows18′.

[0023] Because the capacitance of a multi-plate system varies in aperiodic or cyclic fashion, there will be positions of the rotor forwhich the incremental capacitance change will be zero and no usefulposition information can be detected. A common method to solve thisproblem, illustrated in FIG. 3, is to employee two sets of capacitorplates with an offset between the sets of the plates. In one sucharrangement, the plates are offset such that when one set of plates 20(referred to as Normal or N) reaches a point of minimum sensitivity as afunction of position, the other set of plates 22 (referred to asQuadrature or Q) have maximum sensitivity as a function of position.Conversely, when the Quadrature plates 22 are at a minimum sensitivityas a function of position, the Normal plates 20 have maximum sensitivityas a function of position. This type of arrangement will result inoffset signals N and Q that vary in a periodic or cyclic fashion with alinear change in the position of the rotor in the direction of arrows18″.

[0024] The difficulty in using an arrangement like that in FIG. 3 liesin properly combining the cyclic N and Q signals into a useful linearposition signal representative of the linearly changing position of therotor. A common method of combining the N and Q signals simply selectsone or the other of the signals depending upon which is farthest from apeak or valley in its cyclic variation. As the N and Q signals vary, thesystem switches from one signal to the other. This method suffers fromthe presence of discontinuous behavior at the switch points, which canresult in erroneous determination of the rotor position. Also,corrections for non-linearities in the sensor is difficult to achieve.

[0025] As illustrated in FIG. 4, the system and method for determiningposition employed by one embodiment of the present invention convertsboth the N and Q signals into digital signals, such as by an analog todigital converter (ADC), and then combines the signals. Since the N andQ signals are converted to digital signals, they do not change duringthe process of determining the rotor position. Changes in the N and Qsignals during the position calculation are a significant difficultywith an analog approach to combining the N and Q signals.

[0026] As described in greater detail below, in one embodiment of theinvention, an encoder 30, such as a multi-plate capacitive sensor,generates signals N and Q as a result of movement by a body. The N and Qsignals are converted to digital signals by an analog to digitalconverter 32. The digitized N and Q signals are combined using an NQcombiner table 34, and then a long range position or distance traveledis computed by using a counter 36 to maintain a count of movements ofthe combined signal. A short range (or high resolution) position ordistance traveled is calculated by using linear interpolation from anarctangent look-up table 38, using the value of |N|/|Q| or |Q|/|N|. Thelong range and short range positions are then combined to determine thefinal position of the moving body.

[0027] In one embodiment of the present invention, the N and Q signalscan be viewed as a sine and cosine wave, respectively, so that N=Asin(x) and Q=A cos(x), where x is the linear position of the rotor.Given this knowledge about the shape of the N and Q signals, a linearposition signal may be generated by converting the N and Q signals intothe argument x. This can be done by reversing N/Q=tan(x) to getx=arctan(N/Q).

[0028] Practically, the N and Q signals will not to be perfect sine andcosine waves. To correct for distortions in the shape of the N and Qsignals, a table representing arctan(R) is used, where R=|N|/|Q| when Nis near 0, and R=|Q|/|N| when Q is near 0, where “∥” indicates that theabsolute values of the N and Q signals are used. This ensures that theratio R is well behaved and limits the range required of the divider andthe arctangent table.

[0029]FIG. 5 illustrates a triangle wave representation of the N and Qsignals. Other data regarding the N and Q signals as a function of timeis also shown in FIG. 5: the polarity of N (pN) and the polarity of Q(pQ), where 0 indicates a positive polarity and 1 indicates a negativepolarity; whether |N|>|Q|, where 0 is false and 1 is true; the value ofN/Q; and the value of R, where R=|N|/|Q| when N is near 0, and R=|Q|/|N|when Q is near 0. Selecting the inputs into the R divider in this wayensures that inputs to the digital divider are always positive and thatthe result R ranges from 0 to 1. When an arctangent table is generatedfor this wave shape, the resulting position signal is still linear, aswill be shown below.

[0030] A polar plot of the N and Q signals of FIG. 5 is presented inFIG. 6. As can be seen in FIG. 6, the triangular waveform of the N and Qsignals in FIG. 5 creates a diamond trajectory 40. If N and Q wereperfect sine and cosine waves, the trajectory would appear as the dashedcircle 42 in FIG. 6.

[0031] From FIG. 5, the polarity of N (pN), the polarity of Q (pQ), andwhether |N|>|Q| are known as a function of time for the N and Q signals.From this data, eight unique conditions (see the table in FIG. 7,discussed in greater detail below) can be identified, and it can easilybe determined into which section of the polar plot (FIG. 6) a particularsample NQ (taken at a time t₀ through t_(n)) of the N and Q signalsfalls. Also from FIGS. 5 and 6, it can be seen that an angulardisplacement of 90° (π/2 radians) equates to one half of the plate pitchP. By determining the section (quadrant and/or octal in thisillustration) in which the plot of N and Q falls for each successivesample and then keeping a count of the change in sections, a long range,low resolution position of the rotor can be calculated. For example, ifthe trajectory of N and Q causes the quadrant in which the sample isplotted to successively change four times in the same direction, it canbe determined that the trajectory of the plot moved through an anglebetween 360° and 450° (2π and 5π/2 radians). It can then be calculatedthat the position of the rotor changed between 2P and 5P/2.

[0032] As can be seen, maintaining a count of the change of quadrantsprovides a sensing resolution of P/2. To obtain a higher sensingresolution of P/4, a count of the change in octals could be maintained.For example, if the trajectory of N and Q causes the octal in which thesample is plotted to successively change 6 times in the same direction,it can be determined that the trajectory of the plot moved through anangle between 270° and 315° (3π/2 and 7π/4 radians). It can then becalculated that the position of the rotor changed between 3P/2 and 7P/4.

[0033] However, a sensing resolution equivalent to one quarter of theplate pitch (P/4) may still be lower than that required for manyapplications, such as for use in a data storage device as discussedabove, where a sensing resolution in the range of 0.01 μm to 0.1 μm maybe required. At such high sensing resolutions, when using only a countof the change in octals to determine position, the pitch of the plateswould need to be on the order of 0.04 μm to 0.4 μm. While it may bepossible to produce a plate pitch having these dimensions, doing soreliably is technically difficult and expensive. It would thus bedesirable to obtain a sensing resolution greater than one quarter of theplate pitch.

[0034] The table in FIG. 7, referred to herein as an NQ combiner table,shows the eight unique conditions identifiable using the polarity of N(pN), the polarity of Q (pQ), and whether |N|>|Q|. The eight uniqueconditions correspond to the octal in which the plot of N and Q falls.In the table of FIG. 7, the octals are given count numbers (CntOctal) of0 through 7. The eight octal count numbers likewise correspond to fourquadrant count numbers (CntQuad) of 0 through 3. As illustrated above,by counting the number of times the plot of N and Q moves to a differentoctal or quadrant, the long range position of the rotor may becalculated. To accurately maintain a count of the number of times theplot of N and Q moves to a different octal or quadrant, sampling of Nand Q must occur at a frequency high enough that the plot of N and Qdoes not move more than one quadrant count or octal count (dependingupon which is being counted) between samples

[0035] The table of FIG. 7 shows the case where the long-range positioninformation is generated by maintaining a count of the change in thecount quadrant (CntQuad) from one sample to the next. The sub-quadrantposition (SQ) is generated from the arctangent lookup table (in themanner further described below) and combined with the long-rangeposition to create a linear, high-resolution position signal. It will berecognized that the same method may be employed, using a count of thechange in the octal count from one sample to the next, with appropriatechanges to the values of SQ in Table 1.

[0036] As noted above, R=|N|/|Q| when N is near 0, and R=|Q|/|N| when Qis near 0. Using the value of R designated in Table 1 for particularvalues of pN, pQ and |N|>|Q|, the sub-quadrant position (SQ) may beextracted from the table and is equal to either SQr or P/2−SQr, whereSQr=P arctan(R)/π, where P is the pitch spacing of the capacitiveplates. Because the shape of the N and Q signals will not be perfectlysinusoidal, nor perfectly linear in the case of a triangular waveform,the calculation of the value of arctan(R) may be done with a look-uptable. A look-up table can compensate for irregularities in the shapesof the N and Q signals.

[0037] To limit the number of arctangent table entries that must bestored in the look-up table, a linear interpolation is performed todetermine the SQr value from two adjacent table entries. The value of Rcalculated in Table 1 is truncated until it matches a look-up tableentry R1. R1 is used as the table index and contains SQr1. The nexthigher table index R2 contains SQr2. The linear interpolation calculatesthe value of SQr as:

SQr=(SQr2−SQr1)(R−R1)+SQr1

[0038] where R is scaled so that it ranges over the number of tableentries, i.e., for 128 table entries R ranges from 0 to 128, rather thanfrom 0 to 1. The calculated value of SQr is then used to calculate SQ asdictated by Table 1.

[0039] The arctangent table look-up method will have errors due to thelimited number of table entries, as well as the fixed word widthsavailable in the table contents and the divider. The magnitude of thiserror using linear interpolation in a 128 entry lookup table can beshown to be negligible. For example, for a multi-plate system having apitch (P) and a 128 entry look-up table:

SQra=arctan(R)(radians)

SQr=(P)(SQra)/π(distance)

ΔR=1/127=0.0079

SQra1=arctan(R1)

SQra2=arctan(R1+ΔR)

IntSQra=(SQra1+SQra2)/2

ActSQra=arctan(R1+ΔR/2)

ErrSQra=ActSQra−IntSQra

[0040] where ΔR is the incremental change between successive look-uptable entries, SQra1 is the first table value, SQra2 is the second(adjacent) table value, IntSQra is the midpoint using interpolation,ActSQra is the actual midpoint, and ErrSQra is the error between theactual and interpolated midpoints.

[0041] Using the above equations, a pitch of 5.04 μm and randomlyselecting values of R1 between 0 and 1: ErrSQra Error R1 SQra1 SQra2IntSQra ActSQra (μrad) (pm) 0.5700 0.5181 0.5240 0.5210 0.5210 5.03 8.080.5200 0.4795 0.4857 0.4826 0.4826 5.00 8.02 1.0000 0.7854 0.7893 0.78740.7874 3.86 6.19

[0042] From the table above, it can be seen that using linearinterpolation from a 128 entry table would produce an error ofapproximately 8 picometers (0.008 nm or 8×10⁻⁶ μm) when the capacitiveplates have a pitch of approximately 5 μm. That is, the error isapproximately 1.5/1,000,000 of the pitch.

[0043]FIG. 8 provides a graphic illustrating a simulation using themethod described above and a plate pitch of 5.04 μm. It can be seen thata linear position signal (cx) results, with a very small position error(xerr) that hovers close to zero. It should be noted that the scale oflinear position signal (cx) is in micrometers (μm), while the scale ofposition error (xerr) is in nanometers (nm)

[0044]FIG. 9 shows the same data as FIG. 8, but after the data has beenquantitized using a 12 bit analog to digital conversion, and includinglook-up table errors. Again, a linear position signal (cx) results, withonly a small increase in the position error (xerr).

[0045] A key limitation in creating a useful X-Y position sensor is thevariation in spacing d between capacitor plates. Plate spacingvariations may occur whenever the micromover is commanded to move therotor or when the rotor undergoes external shock and vibration.Variation in the plate spacing causes a corresponding change in thecapacitance of the system. Unfortunately, the capacitance variationcaused by the plate spacing variation is indistinguishable from thedesired area variation, and will thus be misinterpreted as a lateralposition change of the rotor if it is not removed from the detectedsignal.

[0046] In the method described herein, using a small signal analysistechnique, the divide operation done in the NQ combiner cancels theeffect of the gap variation, shown as follows:

For SQ∝Q/N, where Q=K AreaQ(x)/d and N=K AreaN(x)/d,

then SQ∝AreaQ(x)/AreaN(x), as desired, with no effect from gap d,

[0047] where K is the dielectric constant of the material between theplates, AreaN(x) and AreaQ(x) are the overlap areas for the N and Q setsof plates at linear position x, respectively, and d is the plateseparation.

[0048]FIG. 10 is a plot of N and Q signals with a high frequency sinewave modulating the capacitor spacing (such as if the system was subjectto an external vibration). As can be seen from examining FIG. 10, thereis no corresponding modulation in the linear position signal (cx) or inthe position detector error (xerr).

[0049] The invention described herein thus provides a method fordetermining the position of a rotor with respect to a stator whichprovides a linear position signal, has a high sensing resolution as afunction of plate pitch, accounts for irregularities in the inputsignals, and which is insensitive to variations in plate spacing.

[0050] Although specific embodiments have been illustrated and describedherein for purposes of description of the preferred embodiment, it willbe appreciated by those of ordinary skill in the art that a wide varietyof alternate and/or equivalent implementations calculated to achieve thesame purposes may be substituted for the specific embodiments shown anddescribed without departing from the scope of the present invention.Those with skill in the mechanical, electro-mechanical, electrical, andcomputer arts will readily appreciate that the present invention may beimplemented in a very wide variety of embodiments. This application isintended to cover any adaptations or variations of the preferredembodiments discussed herein. Therefore, it is manifestly intended thatthis invention be limited only by the claims and the equivalentsthereof.

What is claimed is:
 1. A method for determining the position of a rotorwith respect to a stator, comprising the steps of: generating offsetNormal (N) and Quadrature (Q) signals when the rotor moves with respectto the stator; sampling the N and Q signals at a predetermined timeinterval; calculating a section of a polar plot in which the N and Qsignal sample lies; counting the change of sections from prior N and Qsignal samples to determine a long range rotor position; calculating asub-section rotor position for the N and Q signal sample; and combiningthe long range rotor position with the sub-section rotor position tocreate a linear position signal indicative of the position of the rotor.2. The method of claim 1, wherein generating the offset N and Q signalscomprises the steps of: providing first and second sets of capacitiveplates having a pitch P on the rotor and stator; and offsetting thefirst and second sets of capacitive plates to generate offset N and Qsignals N and Q, respectively.
 3. The method of claim 2, furthercomprising the step of converting analog N and Q signals to digitalsignals.
 4. The method of claim 2, wherein the first and second sets ofcapacitive plates are offset such that when the first set of platesreaches a point of minimum sensitivity as a function of position, thesecond set of plates have a maximum sensitivity as a function ofposition.
 5. The method of claim 1, wherein the N and Q signals arecyclic as the rotor moves in a linear direction.
 6. The method of claim1, wherein the section in which the N and Q signal sample lies is aquadrant of a polar plot.
 7. The method of claim 1, wherein the sectionin which the N and Q signal sample lies is an octal of a polar plot. 8.The method of claim 1, wherein calculating a section of a polar plot inwhich the N and Q signal sample lies is accomplished using a combinertable.
 9. The method of claim 8, wherein for each N and Q signal sample,the combiner table uses the polarities of the N and Q signals, andwhether |N|>|Q| to determine the section of a polar plot in which the Nand Q signal sample lies.
 10. The method of claim 1, wherein calculatinga sub-section rotor position for the N and Q signal sample isaccomplished using an arctangent look-up table.
 11. The method of claim10, wherein the arctangent look-up table is a 128 entry table.
 12. Themethod of claim 10, wherein the sub-section rotor position is calculatedusing linear interpolation between adjacent arctangent look-up tableentries.
 13. The method of claim 5, wherein the N and Q signals aresinusoidal wave forms.
 14. The method of claim 5, wherein the N and Qsignals are triangular wave forms.
 15. A method for measuring thedistance traveled by a rotor from time t₀ through t_(n), comprising thesteps of: generating offset N and Q signals that vary cyclically withthe rotor position; sampling the N and Q signals at successivepredetermined time intervals t₀ through t_(n) to obtain signal samplesNQt₀ through NQt_(n); converting signal samples NQt₀ through NQt_(n) toa polar coordinate system; determining in which section of the polarcoordinate system each of signal samples NQt₀ through NQt_(n) lies;counting the change of sections from signal samples NQt₀ throughNQt_(n); calculating a long range distance traveled by the rotor usingthe change of sections count; calculating a sub-section rotor distancefor signal sample NQt_(n); and combining the long range rotor distancewith the sub-section rotor distance to measure the distance traveled bythe rotor from time t₀ through t_(n).
 16. The method of claim 15,wherein calculating a sub-section rotor distance for signal sampleNQt_(n) uses the value |N|/|Q| at time t_(n).
 17. The method of claim16, wherein an arctangent look-up table is used to calculate thesub-section rotor distance.
 18. The method of claim 17, wherein linearinterpolation is used to calculate the sub-section rotor distance fromthe arctangent look-up table.
 19. The method of claim 15, wherein thefrequencies of the N and Q signals are proportional to the distancetraveled by the rotor.
 20. The method of claim 15, wherein the frequencyof successive time intervals t₀ through t_(n) is such that the locationof successive samples of the N and Q signals does not change by morethan one section.
 21. The method of claim 15, wherein the section of thepolar coordinate system is a quadrant of the polar coordinate system.22. The method of claim 15, wherein determining in which section of thepolar coordinate system each of signal samples NQt₀ through NQt_(n) liesis accomplished using a combiner table.
 23. The method of claim 22,wherein for each signal sample NQt₀ through NQt_(n), the combiner tableuses the polarities of the N and Q signals, and whether |N|>|Q| todetermine the section of the polar coordinate system in which eachsample lies.
 24. The method of claim 23, wherein the combiner tabledivides the polar coordinate system into eight discrete sections.
 25. Acapacitive position sensor system comprising: an encoder for generatingoffset N and Q signals as a rotor moves with respect to a stator; meansfor converting signal samples NQt₀ through NQt_(n) to a polar coordinatesystem and determining in which section of the polar coordinate systemeach of samples NQt₀ through NQt_(n) lies; means for counting the changeof polar coordinate sections from sample NQt₀ through sample NQt_(n) andcalculating a long range distance traveled by the rotor; means forcalculating a sub-section rotor distance for sample NQt_(n); and meansfor combining the long range rotor distance and the sub-section rotordistance to measure the distance traveled by the rotor from time t₀through t_(n).
 26. The system of claim 25, wherein the means forconverting signal samples NQt₀ through NQt_(n) to a polar coordinatesystem and determining in which section of the polar coordinate systemeach of samples NQt₀ through NQt_(n) lies comprises a combiner tableusing the polarities of the N and Q signals, and whether |N|>|Q|. 27.The system of claim 25, wherein the means for calculating a sub-sectionrotor distance for sample NQt_(n) comprises an arctangent look-up tableusing the value of |N|/|Q| at time t_(n).
 28. The system of claim 27,wherein the means for calculating a sub-section rotor distance using anarctangent look-up table uses linear interpolation between entries inthe look-up table.
 29. The system of claim 25, wherein the N and Qsignals are offset such that when the N signal reaches a point ofminimum sensitivity as a function of position, the Q signal has amaximum sensitivity as a function of position.
 30. The system of claim25, wherein the encoder comprises: a first set of capacitive plateshaving a pitch P on the rotor and the stator for generating an N signalas the rotor moves; and a second set of capacitive plates having a pitchP on the rotor and the stator and offset from the first set ofcapacitive plates for generating a Q signal as the rotor moves.
 31. Thesystem of claim 25, further comprising means for converting analog N andQ signals to digital signals.